Theorem f1ocnvd 7666

Description: Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 30-Apr-2015.)

Hypotheses

Ref	Expression
f1od.1	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)
f1od.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑊)
f1od.3	⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ 𝑋)
f1od.4	⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)))

Assertion

Ref	Expression
f1ocnvd	⊢ (𝜑 → (𝐹:𝐴–1-1-onto→𝐵 ∧ ◡𝐹 = (𝑦 ∈ 𝐵 ↦ 𝐷)))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑦,𝐶   𝑥,𝐷   𝜑,𝑥,𝑦

Allowed substitution hints:   𝐶(𝑥)   𝐷(𝑦)   𝐹(𝑥, 𝑦)   𝑊(𝑥, 𝑦)   𝑋(𝑥, 𝑦)

Proof of Theorem f1ocnvd

Step	Hyp	Ref	Expression
1		f1od.2	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑊)
2	1	ralrimiva 3154	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝑊)
3		f1od.1	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)
4	3	fnmpt 6673	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝑊 → 𝐹 Fn 𝐴)
5	2, 4	syl 18	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴)
6		f1od.3	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ 𝑋)
7	6	ralrimiva 3154	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 𝐷 ∈ 𝑋)
8		eqid 2760	. . . . . 6 ⊢ (𝑦 ∈ 𝐵 ↦ 𝐷) = (𝑦 ∈ 𝐵 ↦ 𝐷)
9	8	fnmpt 6673	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 𝐷 ∈ 𝑋 → (𝑦 ∈ 𝐵 ↦ 𝐷) Fn 𝐵)
10	7, 9	syl 18	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ 𝐷) Fn 𝐵)
11		f1od.4	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)))
12	11	opabbidv 5171	. . . . . 6 ⊢ (𝜑 → {⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} = {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)})
13		df-mpt 5187	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)}
14	3, 13	eqtri 2783	. . . . . . . 8 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)}
15	14	cnveqi 5854	. . . . . . 7 ⊢ ◡𝐹 = ◡{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)}
16		cnvopab 6131	. . . . . . 7 ⊢ ◡{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} = {⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)}
17	15, 16	eqtri 2783	. . . . . 6 ⊢ ◡𝐹 = {⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)}
18		df-mpt 5187	. . . . . 6 ⊢ (𝑦 ∈ 𝐵 ↦ 𝐷) = {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)}
19	12, 17, 18	3eqtr4g 2820	. . . . 5 ⊢ (𝜑 → ◡𝐹 = (𝑦 ∈ 𝐵 ↦ 𝐷))
20	19	fneq1d 6626	. . . 4 ⊢ (𝜑 → (◡𝐹 Fn 𝐵 ↔ (𝑦 ∈ 𝐵 ↦ 𝐷) Fn 𝐵))
21	10, 20	mpbird 260	. . 3 ⊢ (𝜑 → ◡𝐹 Fn 𝐵)
22		dff1o4 6827	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ◡𝐹 Fn 𝐵))
23	5, 21, 22	sylanbrc 595	. 2 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)
24	23, 19	jca 521	1 ⊢ (𝜑 → (𝐹:𝐴–1-1-onto→𝐵 ∧ ◡𝐹 = (𝑦 ∈ 𝐵 ↦ 𝐷)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  {copab 5167   ↦ cmpt 5186  ^◡ccnv 5654   Fn wfn 6528  –1-1-onto→wf1o 6532

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540

This theorem is used by:  f1od  7667  f1ocnv2d  7668  pw2f1ocnv  43881